A positive acute angle is divided into two parts whose tangents are 1 2 and 1 3 . Show that the angle is π 4 .

Let α = A + B According to given data, tan A = 1 2 and tan B = 1 3 Now, tan α = tan ( A + B ) ⇒ tan α = tan A + tan B 1 - tan A tan B ⇒ tan α = 1 2 + 1 3 1 - 1 2 · 1 3 ⇒ tan α = 5 6 5 6 ⇒ tan α = 1 ⇒ α = π 4 Hence, proved that angle is π 4 .

A positive acute angle is divided into two parts whose tangents are 1 2 and 1 3 . Show that the angle is π 4 .

Let α = A + B According to given data, tan A = 1 2 and tan B = 1 3 Now, tan α = tan ( A + B ) ⇒ tan α = tan A + tan B 1 - tan A tan B ⇒ tan α = 1 2 + 1 3 1 - 1 2 · 1 3 ⇒ tan α = 5 6 5 6 ⇒ tan α = 1 ⇒ α = π 4 Hence, proved that angle is π 4 .

If α + β = π 4 , prove that ( 1 + tan α ) ( 1 + tan β ) = 2 .

Given that, α + β = π 4 ⇒ tan ( α + β ) = tan π 4 ⇒ tan α + tan β 1 - tan α tan β = 1 ⇒ tan α + tan β = 1 - tan α tan β ⇒ tan α + tan β + tan α tan β + 1 = 1 + 1 ⇒ ( 1 + tan α ) + tan β ( 1 + tan α ) = 2 ⇒ ( 1 + tan α ) ( 1 + tan β ) = 2 Hence, proved that ( 1 + tan α ) ( 1 + tan β ) = 2 .

Prove that tan 56 ° = cos 11 ° + sin 11 ° cos 11 ° - sin 11 ° .

RHS = cos 11 ° + sin 11 ° cos 11 ° - sin 11 ° = sin 79 ° + sin 11 ° sin 79 ° - sin 11 ° ∵ cos 90 ° - x = sin x We know that, sin A + sin B = 2 sin A + B 2 cos A - B 2 and sin A - sin B = 2 cos A + B 2 sin A - B 2 = 2 sin 45 ° cos 34 ° 2 cos 45 ° sin 34 ° = cot 34 ° = cot 90 ° - 36 ° = tan 56 ° = LHS Hence, proved that tan 56 ° = cos 11 ° + sin 11 ° cos 11 ° - sin 11 °

Prove that 1 + tan x tan x 2 = sec x = tan x cot x 2 - 1 .

Consider, 1 + tan x tan x 2 = 1 + sin x cos x · sin x 2 cos x 2 = cos x cos x 2 + sin x sin x 2 cos x cos x 2 = cos x - x 2 cos x cos x 2 ∵ cos A cos B + sin A sin B = cos ( A - B ) = cos x 2 cos x cos x 2 = 1 cos x ⇒ 1 + tan x tan x 2 = sec x ...(i) Consider, tan x cot x 2 - 1 = sin x cos x · cos x 2 sin x 2 - 1 = sin x · cos x 2 - cos x sin x 2 cos x sin x 2 = sin x - x 2 cos x sin x 2 ∵ sin A cos B - sin B cos A = sin A - B = sin x 2 cos x sin x 2 = 1 cos x ⇒ tan x cot x 2 - 1 = sec x ...(ii) From (i) and (ii), we get 1 + tan x tan x 2 = sec x = tan x cot x 2 - 1 Hence, proved.